Proved. Lim and Steinerberger's Theorem 1 (Mathematika 71 (2025), no. 2, e70009; refereed) gives, for every $c>0$, infinitely many pairs $(m,n)$ with $1\le\sum_{\ell=n}^m1/\ell\le1+c/n^2$; for such a pair the minimal $m(n)$ is at most $m$, so $\epsilon(n)\le c/n^2$, and the pairs have distinct $n$ once $n$ is large (a two-line deduction made here). So $\liminf n^2\epsilon(n)=0$. Their Theorem 2 gives the refined bound $|\sum_{\ell=n}^m1/\ell-1|\le1/(n^2(\log n)^{5/4-\varepsilon})$ for infinitely many pairs; its transfer to $\epsilon(n)$ uses the paper's remark, stated without proof, that the sum can be forced above $1$. The site's label is PROVED (LEAN); its Lean qualifier is a catalog label whose scope is qualified under Formalization and the Lean label below, and no local kernel credit is claimed. The accepted claim is recorded on Lim and Steinerberger's claim page (2024).